Contenido

La teoría de la torsión de Saint-Venant es aplicable a piezas prismáticas de gran inercia torsional con cualquier forma de sección, en esta simplificación se asume que el llamado momento de alabeo es nulo, lo cual no significa que el alabeo seccional también lo sea. La teoría de torsión de Saint-Venant da buenas aproximaciones para valores , esto suele cumplirse en:.

1. • Secciones macizas de gran inercia torsional (circulares o de otra forma).

2. • Secciones tubulares cerradas de pared delgada.

3. • Secciones multicelulares de pared delgada.

Para secciones no circulares y sin simetría de revolución la teoría de Sant-Venant además de un giro relativo de la sección transversal respecto al eje baricéntrico predice un alabeo seccional o curvatura de la sección transversal. La teoría de Coulomb de hecho es un caso particular en el que el alabeo es cero, y por tanto sólo existe giro.

Straight torsion: Coulomb theory

Coulomb's theory is applicable to solid or hollow power transmission shafts; due to the circular symmetry of the section, there cannot be differential warping on the section. According to Coulomb theory, torsion generates a shear stress which is calculated by the formula:

Where:.

This equation is based on Coulomb's kinematic hypothesis on how a prismatic piece with symmetry of revolution deforms, that is, it is a theory applicable only to circular or hollow circular section elements. For pieces with a section of this type, it is assumed that the barycentric axis remains unchanged and any other line parallel to the axis is transformed into a spiral that rotates around the barycentric axis, that is, it is accepted that the deformation is given by displacements of the type:

The strain tensor for a twisted piece like the previous one is obtained by appropriately deriving the previous components of the displacement vector:

From these components of the strain tensor using the Lamé-Hooke equations, the stress tensor is given by:

Using the equivalence equations we arrive at the relationship between the function α and the torque:

Where , is the polar moment of inertia which is the sum of the second moments of area.

Non-straight torsion: Saint-Venant theory

For a straight bar with a non-circular section, in addition to the relative rotation, a small warpage will appear that requires a more complicated kinematic hypothesis. To represent the deformation, a system of axes can be taken in which

Where is the relative rotation of the section (its derivative being constant); being z and y the coordinates of the shear center with respect to the center of gravity of the cross section and ω(y, z) being the unit warping function that gives the displacements perpendicular to the section and allows us to know the final curved shape that the cross section will have. It should be noted that the theory that postulates that the derivative of the spin is constant is only a useful approximation for pieces with great torsional inertia.

By calculating the components of the strain tensor from the derivatives of the displacement, we have:

Calculating the stresses from the previous deformations and introducing them into the elastic equilibrium equation we arrive at:.

Prandtl membrane analogy

For solid sections of great torsional rigidity, the distribution of stresses associated with torsion bears a mechanical analogy with the deformation of a quasi-planar elastic membrane. Specifically, Prandtl proved in 1903 that the shape adopted by the membrane can be related to a stress function whose derivatives give the tangential stresses in each direction. In other words, the slope of a deformed Prandtl membrane coincides with the torsional tangential stresses of a mechanical prism whose cross section has precisely the same shape as the membrane.

In that case for a simply connected (i.e. solid) section, the torsion problem can be posed in terms of the Prandtl stress function which is defined by:.

And in terms of these the tensions are given by:.

Simple thin-walled closed sections

In this case the tangential stresses can be considered approximately constant on a line parallel to the thickness of the piece, that is, perpendicular to the outer contour of the piece. The tangential stress in this case can be expressed by:.

Where:.

While the twist:

In case the thickness is e(s) = econstant, this last equation reduces to:.

Para piezas de muy escasa inercia torsional, como las piezas de pared delgada abierta, puede construirse un conjunto de ecuaciones muy simples en la que casi toda la resistencia a la torsión se debe a las tensiones cortantes inducidas por el alabeo de la sección. En la teoría de torsión alabeada pura se usa la aproximación de que el momento de alabeo coincide con el momento torsor total. Esta teoría se aplica especialmente a piezas de pared delgada abierta, donde no aparecen esfuerzos de membrana.

Thin-walled open sections

For a very elongated rectangle (b << a) the maximum tangential stress and twist can be approximated by:.

For a profile I or profile H that can be approximated by joining rectangles of dimensions (two elongated rectangular wings and an elongated rectangular web) the previous expressions can be generalized to:.

Where τ is the maximum tangential stress on the i-th rectangle, b is the thickness (width) of said rectangle and a its length.

In the domain of dominant Saint-Venant torsion and dominant warped torsion, the Sant-Venant theory and warped torsion theory can be used to a certain degree of approximation. However, in the central domain of extreme torsion, important errors are made and it is necessary to use the more complicated general theory.

Where the geometric magnitudes are respectively the second warping moment and the torsional modulus and the "stresses" are called the bimoment and warping moment, all of them defined for mechanical prisms.

For a piece with a circular section of radius R subjected to a torsional moment M, the maximum tangential stress is given by:.

For an equilateral triangle and a square, the maximum stresses due to the torison occur over half of one of its sides and are given by:.

where L is the side of the triangle or square. For a simply connected general section of great torisonal rigidity subjected to Saint-Venant torsion, and therefore for the Prandtl membrane analogy to be valid, it can be proven that the maximum tangential stress due to torsion is bounded as follows:[3]​.

where:.

• - Twist in plan.

• - Ortiz Berrocal, L., Elasticity, McGraw-Hill, 1998, ISBN 84-481-2046-9.

• - Timoshenko, S.P. and Godier J.N., Theory of elasticity, McGraw-Hill, 1951.

• - Monleón Cremades, S., Analysis of beams, arches, plates and sheets, Ed. UPV, 1999, ISBN 84-7721-769-6.

• - Coulomb shear failure equation.

Contenido

La teoría de la torsión de Saint-Venant es aplicable a piezas prismáticas de gran inercia torsional con cualquier forma de sección, en esta simplificación se asume que el llamado momento de alabeo es nulo, lo cual no significa que el alabeo seccional también lo sea. La teoría de torsión de Saint-Venant da buenas aproximaciones para valores , esto suele cumplirse en:.

1. • Secciones macizas de gran inercia torsional (circulares o de otra forma).

2. • Secciones tubulares cerradas de pared delgada.

3. • Secciones multicelulares de pared delgada.

Para secciones no circulares y sin simetría de revolución la teoría de Sant-Venant además de un giro relativo de la sección transversal respecto al eje baricéntrico predice un alabeo seccional o curvatura de la sección transversal. La teoría de Coulomb de hecho es un caso particular en el que el alabeo es cero, y por tanto sólo existe giro.

Straight torsion: Coulomb theory

Coulomb's theory is applicable to solid or hollow power transmission shafts; due to the circular symmetry of the section, there cannot be differential warping on the section. According to Coulomb theory, torsion generates a shear stress which is calculated by the formula:

Where:.

This equation is based on Coulomb's kinematic hypothesis on how a prismatic piece with symmetry of revolution deforms, that is, it is a theory applicable only to circular or hollow circular section elements. For pieces with a section of this type, it is assumed that the barycentric axis remains unchanged and any other line parallel to the axis is transformed into a spiral that rotates around the barycentric axis, that is, it is accepted that the deformation is given by displacements of the type:

The strain tensor for a twisted piece like the previous one is obtained by appropriately deriving the previous components of the displacement vector:

From these components of the strain tensor using the Lamé-Hooke equations, the stress tensor is given by:

Using the equivalence equations we arrive at the relationship between the function α and the torque:

Where , is the polar moment of inertia which is the sum of the second moments of area.

Non-straight torsion: Saint-Venant theory

For a straight bar with a non-circular section, in addition to the relative rotation, a small warpage will appear that requires a more complicated kinematic hypothesis. To represent the deformation, a system of axes can be taken in which

Where is the relative rotation of the section (its derivative being constant); being z and y the coordinates of the shear center with respect to the center of gravity of the cross section and ω(y, z) being the unit warping function that gives the displacements perpendicular to the section and allows us to know the final curved shape that the cross section will have. It should be noted that the theory that postulates that the derivative of the spin is constant is only a useful approximation for pieces with great torsional inertia.

By calculating the components of the strain tensor from the derivatives of the displacement, we have:

Calculating the stresses from the previous deformations and introducing them into the elastic equilibrium equation we arrive at:.

Prandtl membrane analogy

For solid sections of great torsional rigidity, the distribution of stresses associated with torsion bears a mechanical analogy with the deformation of a quasi-planar elastic membrane. Specifically, Prandtl proved in 1903 that the shape adopted by the membrane can be related to a stress function whose derivatives give the tangential stresses in each direction. In other words, the slope of a deformed Prandtl membrane coincides with the torsional tangential stresses of a mechanical prism whose cross section has precisely the same shape as the membrane.

In that case for a simply connected (i.e. solid) section, the torsion problem can be posed in terms of the Prandtl stress function which is defined by:.

And in terms of these the tensions are given by:.

Simple thin-walled closed sections

In this case the tangential stresses can be considered approximately constant on a line parallel to the thickness of the piece, that is, perpendicular to the outer contour of the piece. The tangential stress in this case can be expressed by:.

Where:.

While the twist:

In case the thickness is e(s) = econstant, this last equation reduces to:.

Para piezas de muy escasa inercia torsional, como las piezas de pared delgada abierta, puede construirse un conjunto de ecuaciones muy simples en la que casi toda la resistencia a la torsión se debe a las tensiones cortantes inducidas por el alabeo de la sección. En la teoría de torsión alabeada pura se usa la aproximación de que el momento de alabeo coincide con el momento torsor total. Esta teoría se aplica especialmente a piezas de pared delgada abierta, donde no aparecen esfuerzos de membrana.

Thin-walled open sections

For a very elongated rectangle (b << a) the maximum tangential stress and twist can be approximated by:.

For a profile I or profile H that can be approximated by joining rectangles of dimensions (two elongated rectangular wings and an elongated rectangular web) the previous expressions can be generalized to:.

Where τ is the maximum tangential stress on the i-th rectangle, b is the thickness (width) of said rectangle and a its length.

In the domain of dominant Saint-Venant torsion and dominant warped torsion, the Sant-Venant theory and warped torsion theory can be used to a certain degree of approximation. However, in the central domain of extreme torsion, important errors are made and it is necessary to use the more complicated general theory.

Where the geometric magnitudes are respectively the second warping moment and the torsional modulus and the "stresses" are called the bimoment and warping moment, all of them defined for mechanical prisms.

For a piece with a circular section of radius R subjected to a torsional moment M, the maximum tangential stress is given by:.

For an equilateral triangle and a square, the maximum stresses due to the torison occur over half of one of its sides and are given by:.

where L is the side of the triangle or square. For a simply connected general section of great torisonal rigidity subjected to Saint-Venant torsion, and therefore for the Prandtl membrane analogy to be valid, it can be proven that the maximum tangential stress due to torsion is bounded as follows:[3]​.

where:.

• - Twist in plan.

• - Ortiz Berrocal, L., Elasticity, McGraw-Hill, 1998, ISBN 84-481-2046-9.

• - Timoshenko, S.P. and Godier J.N., Theory of elasticity, McGraw-Hill, 1951.

• - Monleón Cremades, S., Analysis of beams, arches, plates and sheets, Ed. UPV, 1999, ISBN 84-7721-769-6.

• - Coulomb shear failure equation.